Second-Order/Pseudo-Second-Order Reaction
Gary L. Bertrand
Professor Emeritus of Chemistry
Missouri University of Science and Technology
(formerly University of Missouri-Rolla)

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A Second-Order Reaction is described mathematically as:

1/[A]t = 1/[A]t=0 + νAkt

For a Pseudo-Second-Order Reaction, the reaction rate constant k is replaced by the apparent reaction rate constant k'. If the reaction is not written out specifically to show a value of νA, the value is assumed to be 1 and is not shown in these equations.

The simplest way to confirm that these are the proper equations to describe the reaction and to get an approximate value for k or k', is with the method of half-lives. The half-life t1/2 is defined as the time required for the initial concentration to be halved:

[A]t1/2 = [A]t=0/2 .
At this point in time,
2/[A]t=0 - 1/[A]t=0 = 1/[A]t=0 = νAkt1/2

The time required for the concentration to be reduced to 1/4 of the initial concentration (t3/4) is

4/[A]t=0 - 1/[A]t=0 = 3/[A]t=0 = νAkt3/4

NOTE:The "Second-Order Kinetics" exercise is designed so that you can get an accurate value of the half-life. Scan the concentration column for a value slightly larger than half of the initial concentration. Adjust the value of "Start Time" and the value of "Interval" to get a better estimate of the half-life. The same procedure may be used to find the time for the second half-life. A better procedure is to generate data for a much different initial concentration (perhaps a ten-fold change) and confirm that the product of the half-life times the initial concentration is constant and equal to 1/k.

The best values of the reaction rate constant (k) can be obtained with data taken in the middle third of the reaction (from [A]t = (2/3)[A]t=0 to [A]t = (1/3)[A]t=0). Linear Least Squares regression with Y = 1/[A]t and X = t gives k or k' as the slope.